Graham's Number
Not a problem. Just for interest.
If you thought you knew what a big number is, then you may be in
for a surprise and a headache. Follow this link:
http://en.wikipedia.org/wiki/Graham%27s_number
If you thought you knew what a big number is, then you may be in
for a surprise and a headache. Follow this link:
http://en.wikipedia.org/wiki/Graham%27s_number
Labels: friday special, SharedPuzzle
posted by Chris at
7:18 PM
17 Comments:
Thanks Ragknot, I had only just noticed the problem. First go broke the page, that's why your post has disappeared.
hehe thank you for that one. Already knew about grahams number but that brought me to get more into that knuth-arrow-notation which i hadn't understood very well. I just say "crazy shit!!"
=)
I don't know if it mentioned it in the article, but I somewhere read that professional mathematicians have said the number was bigger than their idea of infinity. Of course they really know that conpared to infinity, Graham's number is nothing.
If it makes any of you feel better, I get lost during expansion of 3!!!3, yet alone 3!!!!3 = g1. From there on it gets progressively more insane in a progressively more insane in a progressively more insane in a progressively more insane etc. etc. and more and more insanely etc. bigger way. etc. etc.
... and that does even begin scratch at the surface of how to begin to give you an idea of how insanely big this number is.
... yet it is most definitely a finite number.
okay the 3!!!3 is still possible, maybe try it with smaller numbers (like two or so ^^) and take your time. Its not easy! But what comes later....too brutal for everything.
In the "chained arrow notation, which is way harder than the 3!!!... u have to use three arrows MUHAHA...
here are two other brief articles about it, the first one helped me to get an idea:
http://www-users.cs.york.ac.uk/susan/cyc/g/graham.htm
http://yudkowsky.net/obsolete/singularity.html#beyond_big
Hmmm, it seems that the Knuth notation is defined slightly differently from what had supposed.
If I had invented it, I would certainly not have defined the redundant a!b = a^b, I would have defined a!b = a^(a^(a^...))) with a power tower of height b. I say this without checking that it doesn't make anything ugly.
So I can get a step further than I thought. 3!!3 = 3^(3^3)) = 3^27 = 7.625597484987*10^12. Then 3!!!3 = 3!!(3!!3) = 3^(3^(3^(....))) with a tower height of 7.6*10^12 (approx), the total value of which is beyond my comprehension.
So I can't hope to write 3!!!!3 (=g1) using a simple power operator.
This post has been removed by a blog administrator.
Hehe, and i forgot that little nice sentence in the wikipedia article:
"Even the mere number of towers in this formula for g1 is far greater than the number of Planck volumes into which one can imagine subdividing the observable universe."
Have fun thinking about
... I haven't nipped orft to see what those even bigger numbers are about. Perhaps we should use the silent "b" when talking of these suckers.
Ooooh, I almost forgot. The late Paul Cohen has given me hope. Before him, everyone said that Aleph[1] was the same cardinality as the continuum "c". I never liked that as aleph[1] always struck me as been based on a granular structure, whereas the continuum is the epitome of smoothness (in my mind). Cohen seems to have paved the way to showing that c is the highest cardinality of all the infinities (thought up so far).
Hated even more that aleph[2+]>c somehow implied the existence of something smoother than smooth. On the other hand, I also liked the idea of smoother than smooth, sounded cool.
Greetz
I think I've understood why Knuth used the double arrow as the new operator, and the single up arrow as the usual power operation. It's because of the difficulty of typing the up arrow symbol especially in plain text format.
My reasoning is slightly back to front. If you can't type an up arrow use ^ instead.
Then a^b same as ever, a^^b, a^^^b etc. for the new stuff. Leaves me with the problem of why didn't he use ^ in the first place.
My number is even bigger than the Graham number. It's so big I don't even feel like typing it out.
If I did type it out it would take over 9,999,999,999,999,999,999,999,999,999 life times to type out at maximum speed.
Graham's number cannot ever be typed out in full. Even the first step (g1) cannot be typed out in full. If all the material in the universe was available, it wouldn't let you make the slighest dent in the amount of typing that was required.
How about this for an even bigger number. In constructing Graham's number, we have g1, g2, and so on, up to g64. Now consider g65, g66, and so on, up to g(Graham's number)!! Call this number a. Then consider g(a); call this b; consider g(b), and so on....
You can go on forever. The point was that Graham's number (until recently) was the largest finite number that was the consequence of a "serious" mathematical problem.
The Knuth up-arrow, is not remotely powerful enough to deal with really big numbers though ;)
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